Our algorithm uses the layer approximation method proposed by Indyk and Woodruff in [1]} and generalized to symmetric norms in [2]}, [3]}.
Since symmetric norms are invariant under reordering of the coordinates, the main idea in [1]} is to construct a “layer vector” as follows: for a vector \(v\in \mathbb {R}^d\) , round (the absolute value of) each coordinate to the nearest power \(\alpha ^i\)  for some fixed \(\alpha \in \mathbb {R}\)  and \(i\in \mathbb {N}\) , and then sort the coordinates in an increasing order. This ensures that the \(i\) -th layer contains all the coordinates \(j\in [d]\)  satisfying: \(\alpha ^{i - 1} < |v_j| \le \alpha ^i\) .
In particular, the layer vector of \(v\)  has the form
\(\mathcal {L}(v) := (\underbrace{\alpha ^1, \dots , \alpha ^1}_{b_1~\text{times}},~ \underbrace{\alpha ^2,\dots ,\alpha ^2}_{b_2~\text{times}},~ \cdots , \underbrace{\alpha ^P,\dots ,\alpha ^P}_{b_P~\text{times}},~ 0, \dots , 0) \in \mathbb {R}^d,\) 
